题目
A researcher is looking into the effectiveness of a new medicine for the relief of symptoms. He collects random samples of 8 people who are taking the medicine from each of 50 different medical practices. The number of people who say that the medicine is a success, in each sample, is recorded. The results are summarised in the table below.
| Number of successes | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | | --- | --- | --- | --- | --- | --- | --- | --- | --- | | Number of practices | 4 | 6 | 3 | 12 | 10 | 7 | 4 | 2 | 2 |
The researcher decides to model this data using a binomial distribution.
(a) State two necessary assumptions that the researcher made in order to use this model.
(b) Show that the mean number of successes per sample is 3.54
He decides to use this mean to calculate expected frequencies. The results are shown in the table below.
| Number of successes | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|---|
| Expected frequency | 0.47 | 2.96 | 8.23 | 13.07 | f | 8.23 | 3.27 | 0.74 | g |
(c) Calculate the value of f and the value of g. Give your answers to 2 decimal places.
(d) Stating your hypotheses clearly, test at the 10% level of significance, whether or not the binomial distribution is a suitable model for the number of successes in samples of 8 people.
题目中文翻译
一位研究者正在研究一种新药缓解症状的效果。他从 50 家不同的医疗机构中，每家随机抽取 8 名服药者组成样本，并记录每个样本中认为药物有效的人数。结果如下表。
| 成功次数 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|---|
| 医疗机构数 | 4 | 6 | 3 | 12 | 10 | 7 | 4 | 2 | 2 |
研究者决定用二项分布来建模这些数据。
(a) 写出研究者为了使用这个模型所做的两条必要假设。
(b) 证明每个样本中的平均成功次数为 3.54。
他决定用这个均值来计算期望频数。结果如下表。
| 成功次数 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|---|
| 期望频数 | 0.47 | 2.96 | 8.23 | 13.07 | f | 8.23 | 3.27 | 0.74 | g |
(c) 求 f 和 g 的值。答案保留 2 位小数。
(d) 清楚写出假设，在 10% 显著性水平下检验：二项分布是否适合作为每组 8 人样本中成功人数的模型。